I am interested in expressions of the form

((A → B) → C)

Specifically, given A, I would like to infer C.

I have found that

(((A → B) ∧ A) → (((A → B) → C) → C))

is a valid argument. But maybe I am headed off in the wrong direction. What I am trying to get to is a consequent of just C. Can I get there from here? How?

Thanks